#119772

order	512
nilpotency class	3
isoclinism	isoclinic class of #119765
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(3,7,4,12)(5,10)(6,9)(8,13,16,11)(14,15)(17,18,21,26,22,24,20,25)(19,23,28,30,29,31,27,32);(1,3,5,4)(2,7,10,12)(6,13,14,11)(8,9,16,15)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,32)(26,30);(1,4,5,3)(2,8)(6,13,14,11)(7,15)(9,12)(10,16)(18,24)(19,27)(20,21)(23,30)(28,29)(31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,e^b=eh,f^b=fi,g^b=gh,f^c=fi,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26968);